Four-dimensional De Sitter Space Research Articles

What if string theory has a de Sitter excited state?

We propose precise effective field theory criteria to obtain a four-dimensional de Sitter space within M-theory. To this effect, starting with the state space described by the action of metric perturbations, fluxes etc over the supersymmetric Minkowski vacuum in eleven-dimensions, we discuss the most general low energy effective action in terms of the eleven-dimensional fields including non-perturbative and non-local terms. Given this, our criteria to obtain a valid four-dimensional de Sitter solution at far IR involve satisfying the Schwinger-Dyson equations of the associated path integral, as well as obeying positivity constraints on the dual IIA string coupling and its time derivative. For excited states, the Schwinger-Dyson equations imply an effective emergent potential different from the original potential. We show that while vacuum solutions and arbitrary coherent states fail to satisfy these criteria, a specific class of excited states called the Glauber-Sudarshan states obey them. Using the resurgent structure of observables computed using the path integral over the Glauber-Sudarshan states, four-dimensional de Sitter in the flat slicing can be constructed using a Glauber-Sudarshan state in M-theory.Among other novel results, we discuss the smallness of the positive cosmological constant, including the curious case where the cosmological constant is very slowly varying with time. We also discuss the resolution of identity with the Glauber-Sudarshan states, generation and the convergence properties of the non-perturbative and the non-local effects, the problems with the static patch and other related topics. We analyze briefly the issues related to the compatibility of the Wilsonian effective action with Borel resummations and discuss how they influence the effective field theory description in a four-dimensional de Sitter space.

Wilson Loops at Large N and the Quantum M2-Brane.

The Wilson loop operator in the U(N)_{k}×U(N)_{-k} Aharony-Bergman-Jafferis-Maldacena theory at large N and fixed level k has a dual description in terms of a wrapped M2-brane in the M-theory given by the product of four-dimensional anti de Sitter space (AdS_{4}) and S^{7}/Z_{k}. We consider the localization result for the 1/2-Bogomol'nyi-Prasad-Sommerfield circular Wilson loop expectation value W in this regime and compare it to the prediction of the M2-brane theory. The leading large N exponential factor is matched as expected by the classical action of the M2-brane solution with AdS_{2}×S^{1} geometry. We show that the subleading k-dependent prefactor in W is also exactly reproduced by the one-loop term in the partition function of the wrapped M2-brane (with all Kaluza-Klein modes included). This appears to be the first case of an exact matching of the overall numerical prefactor in the Wilson loop expectation value against the dual holographic result. It provides an example of a consistent quantum M2-brane computation, suggesting various generalizations.

A low-energy limit of Yang-Mills theory on de Sitter space

We consider Yang-Mills theory with a compact structure group G on four-dimensional de Sitter space dS4. Using conformal invariance, we transform the theory from dS4 to the finite cylinder mathcal{I} × S3, where mathcal{I} = (−π/2, π/2) and S3 is the round three-sphere. By considering only bundles P → mathcal{I} × S3 which are framed over the temporal boundary ∂ mathcal{I} × S3, we introduce additional degrees of freedom which restrict gauge transformations to be identity on ∂ mathcal{I} × S3. We study the consequences of the framing on the variation of the action, and on the Yang-Mills equations. This allows for an infinite-dimensional moduli space of Yang-Mills vacua on dS4. We show that, in the low-energy limit, when momentum along mathcal{I} is much smaller than along S3, the Yang-Mills dynamics in dS4 is approximated by geodesic motion in the infinite-dimensional space mathcal{M} vac of gauge-inequivalent Yang-Mills vacua on S3. Since mathcal{M} vac ≅ C∞(S3, G)/G is a group manifold, the dynamics is expected to be integrable.

Four-dimensional de Sitter space is a Glauber-Sudarshan state in string theory

We show that four-dimensional de Sitter space is a Glauber-Sudarshan state, i.e. a coherent state, over a supersymmetric solitonic background in full string theory. We argue that such a state is only realized in the presence of temporally varying degrees of freedom and after including quantum corrections, with supersymmetry being broken spontaneously. On the other hand, fluctuations over the resulting de Sitter space is governed by the Agarwal-Tara state, which is a graviton (and flux)-added coherent state. Once de Sitter space is realized as a coherent state, and not as a vacuum, its ability to remain out of the swampland as well as issues regarding its (meta)stability, vacuum energy, and finite entropy appear to have clear resolutions.

Strong-coupling dynamics and entanglement in de Sitter space

We use holography to study the dynamics of a strongly-coupled gauge theory in four-dimensional de Sitter space with Hubble rate H. The gauge theory is non-conformal with a characteristic mass scale M. We solve Einstein’s equations numerically and determine the time evolution of homogeneous gauge theory states. If their initial energy density is high compared with H4 then the early-time evolution is well described by viscous hydrodynamics with a non-zero bulk viscosity. At late times the dynamics is always far from equilibrium. The asymptotic late-time state preserves the full de Sitter symmetry group and its dual geometry is a domain-wall in AdS5. The approach to this state is characterised by an emergent relation of the form mathcal{P} = w ℰ that is different from the equilibrium equation of state in flat space. The constant w does not depend on the initial conditions but only on H/M and is negative if the ratio H/M is close to unity. The event and the apparent horizons of the late-time solution do not coincide with one another, reflecting its non-equilibrium nature. In between them lies an “entanglement horizon” that cannot be penetrated by extremal surfaces anchored at the boundary, which we use to compute the entanglement entropy of boundary regions. If the entangling region equals the observable universe then the extremal surface coincides with a bulk cosmological horizon that just touches the event horizon, while for larger regions the extremal surface probes behind the event horizon.

De Sitter space as a Glauber-Sudarshan state

Glauber-Sudarshan states, sometimes simply referred to as Glauber states, or alternatively as coherent and squeezed-coherent states, are interesting states in the configuration spaces of any quantum field theories, that closely resemble classical trajectories in space-time. In this paper, we identify four-dimensional de Sitter space as a coherent state over a supersymmetric Minkowski vacuum. Although such an identification is not new, what is new however is the claim that this is realizable in full string theory, but only in conjunction with temporally varying degrees of freedom and quantum corrections resulting from them. Furthermore, fluctuations over the de Sitter space is governed by a generalized graviton (and flux)-added coherent state, also known as the Agarwal-Tara state. The realization of de Sitter space as a state, and not as a vacuum, resolves many issues associated with its entropy, zero-point energy and trans-Planckian censorship, amongst other things.

From Yang–Mills in de Sitter Space to Electromagnetic Knots

We review analytic SU(2) Yang–Mills solutions with finite action on four-dimensional de Sitter space from a new perspective, by conformally mapping dS $$_{4}$$ to a finite Lorentzian cylinder $$(0,\pi ) \times {{S}^{3}}$$ . As a byproduct, all abelian (i.e. Maxwell) solutions are classified by SO(4) representations. Conformal equivalence of (two copies of half of) this cylinder to Minkowski space yields a complete set of rational Maxwell solutions on the latter, which are known as electromagnetic knots. Their properties are efficiently computed on de Sitter space. We close with a couple of explicit examples.

De Sitter field equations from quadratic curvature gravity: A group theoretical approach

In this paper, the linearized field equations related to the quadratic curvature gravity theory have been obtained in the four-dimensional de Sitter (dS) space–time. The massless spin-2 field equations have been written in terms of the Casimir operators of dS group making use of the ambient space notations. By imposing some simple constraints, arisen from group theoretical interpretation of the field equations, a new four-dimensional Gauss–Bonnet (GB-)like action has been introduced with the related field equations transforming according to the unitary irreducible representations (UIRs) of dS group. Since, the field equations transform according to the UIRs of dS group, the GB-like action, we just obtained, is expected to be a successful model of modified gravity. For more clarity, the gauge invariant field equations have been solved in terms of a gauge-fixing parameter [Formula: see text]. It has been shown that the solution can be written as the multiplication of a symmetric rank-2 polarization tensor and a massless minimally coupled scalar field on dS space. The Krein–Gupta–Bleuler quantization method has been utilized and the covariant two-point function has been calculated in terms of the massless minimally coupled scalar two-point function, using the ambient space notations. It has been written in terms of dS intrinsic coordinates from the ambient space counterpart. The two-point functions are dS invariant and free of any theoretical problems. It means that the proposed model is a successful model of modified gravity and it can produce significant results in the contexts of classical theory of gravity and quantum gravity toy models.

Spinors in supersymmetric dS/CFT

We study fermionic bulk fields in the dS/CFT dualities relating mathcal{N} = 2 su- persymmetric Euclidean vector models with reversed spin-statistics in three dimensions to supersymmetric Vasiliev theories in four-dimensional de Sitter space. These dualities specify the Hartle-Hawking wave function in terms of the partition function of deforma- tions of the vector models. We evaluate this wave function in homogeneous minisuperspace models consisting of supersymmetry-breaking combinations of a half-integer spin field with either a scalar, a pseudoscalar or a metric squashing. The wave function appears to be well-behaved and globally peaked at or near the supersymmetric de Sitter vacuum, with a low amplitude for large deformations. Its behavior in the semiclassical limit qualitatively agrees with earlier bulk computations both for massless and massive fermionic fields.

On the dynamics of near-extremal black holes

We analyse the dynamics of near-extremal Reissner-Nordström black holes in asymptotically four-dimensional Anti de Sitter space (AdS4). We work in the spherically symmetric approximation and study the thermodynamics and the response to a probe scalar field. We find that the behaviour of the system, at low energies and to leading order in our approximations, is well described by the Jackiw-Teitelboim (JT) model of gravity. In fact, this behaviour can be understood from symmetry considerations and arises due to the breaking of time reparametrisation invariance. The JT model has been analysed in considerable detail recently and related to the behaviour of the SYK model. Our results indicate that features in these models which arise from symmetry considerations alone are more general and present quite universally in near-extremal black holes.

Pure Yang–Mills Solutions on dS4

We consider pure SU(2) Yang–Mills theory on four-dimensional de Sitter space dS4 and construct smooth and spatially homogeneous classical Yang–Mills fields. Slicing dS4 as $$\mathbb{R} \times {{S}^{3}},$$ via an SU(2)-equivariant ansatz we reduce the Yang–Mills equations to ordinary matrix differential equations and further to Newtonian dynamics in a particular three-dimensional potential. Its classical trajectories yield spatially homogeneous Yang–Mills solutions in a very simple explicit form, depending only on de Sitter time with an exponential decay in the past and future. These configurations have not only finite energy, but their action is also finite and bounded from below. We present explicit coordinate representations of the simplest examples (for the fundamental SU(2) representation). Instantons (Yang–Mills solutions on the Wick-rotated $${{S}^{4}}$$ ) and solutions on AdS4 are also briefly discussed.

Supersymmetric dS/CFT

We put forward new explicit realisations of dS/CFT that relate mathcal{N} = 2 supersymmetric Euclidean vector models with reversed spin-statistics in three dimensions to specific supersymmetric Vasiliev theories in four-dimensional de Sitter space. The partition function of the free supersymmetric vector model deformed by a range of low spin deformations that preserve supersymmetry appears to specify a well-defined wave function with asymptotic de Sitter boundary conditions in the bulk. In particular we find the wave function is globally peaked at undeformed de Sitter space, with a low amplitude for strong deformations. This suggests that supersymmetric de Sitter space is stable in higher-spin gravity and in particular free from ghosts. We speculate this is a limiting case of the de Sitter realizations in exotic string theories.

Solutions to Yang-Mills Equations on Four-Dimensional de Sitter Space.

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter space dS_{4} and construct a smooth and spatially homogeneous magnetic solution to the Yang-Mills equations. Slicing dS_{4} as R×S^{3}, via an SU(2)-equivariant ansatz, we reduce the Yang-Mills equations to ordinary matrix differential equations and further to Newtonian dynamics in a double-well potential. Its local maximum yields a Yang-Mills solution whose color-magnetic field at time τ∈R is given by B[over ˜]_{a}=-1/2I_{a}/(R^{2}cosh^{2}τ), where I_{a} for a=1, 2, 3 are the SU(2) generators and R is the de Sitter radius. At any moment, this spatially homogeneous configuration has finite energy, but its action is also finite and of the value -1/2j(j+1)(2j+1)π^{3} in a spin-j representation. Similarly, the double-well bounce produces a family of homogeneous finite-action electric-magnetic solutions with the same energy. There is a continuum of other solutions whose energy and action extend down to zero.

Higher spin realization of the DS/CFT correspondence

We conjecture that Vasiliev’s theory of higher spin gravity in four-dimensional de Sitter space (dS4) is holographically dual to a three-dimensional conformal field theory (CFT3) living on the spacelike boundary of dS4 at future timelike infinity. The CFT3 is the Euclidean Sp(N) vector model with anticommuting scalars. The free CFT3 flows under a double-trace deformation to an interacting CFT3 in the IR. We argue that both CFTs are dual to Vasiliev dS4 gravity but with different future boundary conditions on the bulk scalar field. Our analysis rests heavily on analytic continuations of bulk and boundary correlators in the proposed duality relating the O(N) model with Vasiliev gravity in AdS4.

Noncommutative de Sitter and FRW spaces

Several versions of fuzzy four-dimensional de Sitter space are constructed using the noncommutative frame formalism. Although all noncommutative spacetimes which are found have commutative de Sitter metric as a classical limit, the algebras and the differential calculi which define them have many differences, which we derive and discuss.

Holography for a non-inflationary early universe

We construct a gravitational dual of the pseudo-conformal universe, a proposed alternative to inflation in which a conformal field theory in nearly flat space develops a time dependent vacuum expectation value. Constructing this dual amounts to finding five-dimensional domain-wall spacetimes with anti-de Sitter asymptotics, for which the wall has the symmetries of four-dimensional de Sitter space. This holographically realizes the characteristic symmetry breaking pattern O(2,4) to O(1,4) of the pseudo-conformal universe. We present an explicit example with a massless scalar field, using holographic renormalization to obtain general expressions for the renormalized scalar and stress-tensor one-point functions. We discuss the relationship between these solutions and those of four-dimensional holographic defect conformal field theories which break O(2,4) to O(2,3)

Schwinger effect in 4D de Sitter space and constraints on magnetogenesis in the early universe

We investigate pair creation by an electric field in four-dimensional de Sitter space. The expectation value of the induced current is computed, using the method of adiabatic regularization. Under strong electric fields the behavior of the current is similar to that in flat space, while under weak electric fields the current becomes inversely proportional to the mass squared of the charged field. Thus we find that the de Sitter space obtains a large conductivity under weak electric fields in the presence of a charged field with a tiny mass. We then apply the results to constrain electromagnetic fields in the early universe. In particular, we study cosmological scenarios for generating large-scale magnetic fields during the inflationary era. Electric fields generated along with the magnetic fields can induce sufficiently large conductivity to terminate the phase of magnetogenesis. For inflationary magnetogenesis models with a modified Maxwell kinetic term, the generated magnetic fields cannot exceed 10^{-30} G on Mpc scales in the present epoch, when a charged field carrying an elementary charge with mass of order the Hubble scale or smaller exists in the Lagrangian. Similar constraints from the Schwinger effect apply for other magnetogenesis mechanisms.

Incompressible fluids of the de Sitter horizon and beyond

There are (at least) two surfaces of particular interest in eternal de Sitter space. One is the timelike hypersurface constituting the lab wall of a static patch observer and the other is the future boundary of global de Sitter space. We study both linear and non-linear deformations of four-dimensional de Sitter space which obey the Einstein equation. Our deformations leave the induced conformal metric and trace of the extrinsic curvature unchanged for a fixed hypersurface. This hypersurface is either timelike within the static patch or spacelike in the future diamond. We require the deformations to be regular at the future horizon of the static patch observer. For linearized perturbations in the future diamond, this corresponds to imposing incoming flux solely from the future horizon of a single static patch observer. When the slices are arbitrarily close to the cosmological horizon, the finite deformations are characterized by solutions to the incompressible Navier-Stokes equation for both spacelike and timelike hypersurfaces. We then study, at the level of linearized gravity, the change in the discrete dispersion relation as we push the timelike hypersurface toward the worldline of the static patch. Finally, we study the spectrum of linearized solutions as the spacelike slices are pushed to future infinity and relate our calculations to analogous ones in the context of massless topological black holes in AdS$_4$.

PROBING THE VACUUM STRUCTURE OF SPACETIME

We explore the question of how to probe the vacuum structure of space time by a massive scalar field through interaction with background gravitons. Using the Γ-regularization for the in-/out-state formalism, we find the effective action of a scalar field in a conformally, asymptotically flat spacetime and a four-dimensional de Sitter space, which is a gravitational analog of the Heisenberg-Euler and Schwinger effective action for a charged scalar in a constant electric field. The effective action is nonperturbative in that it sums all one-loop diagrams with arbitrary number of external lines of gravitons. The massive scalar field becomes unstable due to particle production, the effective action has an imaginary part that determines the decay rate of the vacuum, and the out-vacuum is unitarily inequivalent to the in-vacuum.

Enhanced instability of de Sitter space in Einstein-Gauss-Bonnet gravity

We show that the addition of a topological Gauss-Bonnet term to the gravitational action can greatly increase the instability of four-dimensional de Sitter space, by favoring the nucleation of black holes. The pair-production rate given by the Euclidean action for the instanton takes the form exp(Delta S) where S is the entropy in Einstein-Gauss-Bonnet theory. The coefficient of the Gauss-Bonnet term in the action sets a stability bound on the curvature of empty de Sitter space. For that coefficient in the low-energy effective action of heterotic string theory, the maximal curvature of de Sitter space is in general much lower than the Planck scale.